Un-modelled Pioneer 10/11 clock acceleration from the Schwarzschild metric and Flyby energy Increase

Abstract

When a space-craft reaches a large radial velocity, we expand the time dependent radial distance to linear order in time. In the Schwarzschild metric this linear term in time leads to a significant un-modelled clock acceleration. For Pioneer 10, this acceleration is found to be about $40\%$ of the observed anomalous acceleration. As measured distances are geocentric distances instead of the heliocentric distances in the model, this un-modelled acceleration has an annual oscillation with amplitude close to observations. For Pioneer 11, we find that the un-modelled acceleration "On-Set" is caused by the flyby energy Increase to the space-craft's orbit.